Warped Product CR-submanifolds in locally conformal kaeler manifolds

Recently, B. Y. Chen introduced the notion of warped product CR-submanifolds
and CR-warped products of Kaehler manifolds, that is, a warped product
Riemannian submanifold of a holomorphic submanifold and a totally real submanifold
in a Kaehler manifold ([C-3]). [C-3] follows [MM], [Mi]. In this paper we find
a lot of essential and interesting properties of these submanifolds.
In this paper, we research such submanifolds in locally conformal Kaehler
manifolds.
There are two types of warped product CR-submanifolds. But, one of them is
not interest to us, as it becomes trivial under a certain condition. (See Theorem 2.2).
So, we shall concentrate on another type (we call it a CR-warped product). In a CRwarped
product in an l.c.K.-manifold, we prove one inequality (See Theorem 4.1).
Next, we consider the equality case and we show that some anti-holomorphic CRwarped
product satisfying a certain condition in an l.c.K.-manifold satisfy the equality
(See Theorem 4.3). Finally, in a proper CR-warped product which satisfies the
equality, we prove that its holomorphic submanifold in an l.c.K.-space form is also
an l.c.K.-space form and its totally real submanifold is a real space form (See Theorems
4.6 and 4.7).